Slot Machine with a re-roll option

[probably]

In my math class, me and some friends are making a slot machine for a big project. One of the options for our slot machine gives the spinner extra spins if you get it. We are wondering how to calculate this into our average return per spin, as there is a possibility of getting multiple time machine options (what gives extra spins) in a row, and each extra spin also adds to the average return.

Thanks,
-Greg

 

[HallsofIvy]

I really don't understand what you are saying here. Under what conditions is there a "reroll" and what is the probability of a "reroll"?